mathematics Article Dealing with Degeneracies in Automated Theorem Proving in Geometry Zoltán Kovács 1,† , Tomas Recio 2,*,† , Luis F. Tabera 3,† and M. Pilar Vélez 2,† 1 The Private University College of Education of the Diocese of Linz, Salesianumweg 3, 4020 Linz, Austria; [email protected] 2 Departamento de Ingeniería Industrial, Escuela Politécnica Superior, Universidad Antonio de Nebrija, C/Pirineos 55, 28040 Madrid, Spain; [email protected] 3 Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avenida de los Castros, 39071 Santander, Spain; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Abstract: We report, through different examples, the current development in GeoGebra, a widespread Dynamic Geometry software, of geometric automated reasoning tools by means of computational algebraic geometry algorithms. Then we introduce and analyze the case of the degeneracy conditions that so often arise in the automated deduction in geometry context, proposing two different ways for dealing with them. One is working with the saturation of the hypotheses ideal with respect to the ring of geometrically independent variables, as a way to globally handle the statement over all non-degenerate components. The second is considering the reformulation of the given hypotheses ideal—considering the independent variables as invertible parameters—and developing and exploiting the specific properties of this zero-dimensional case to analyze individually the truth of the statement over the different non-degenerate components. Citation: Kovács, Z.; Recio, T.; Tabera, L.F.; Vélez, M.P. Dealing with Keywords: automated theorem proving in geometry; automated deduction in geometry; automated Degeneracies in Automated Theorem reasoning in geometry; Dynamic Geometry; GeoGebra; computational algebraic geometry Proving in Geometry. Mathematics 2021, 9, 1964. https://doi.org/ 10.3390/math9161964 1. Introduction Academic Editor: Eugenio This paper deals with some issues related to the successful approach to the automated Roanes-Lozano verification and derivation of theorems in elementary geometry through computational algebraic geometry techniques, initiated more than forty years ago by Wu [1]. Received: 29 June 2021 The relatively recent implementation of geometric automated reasoning tools (GART) Accepted: 13 August 2021 Published: 17 August 2021 on Dynamic Geometry programs (See https://en.wikipedia.org/wiki/List_of_interactive_ geometry_software, accessed on 13 August 2021, for a table providing a list of the different Publisher’s Note: MDPI stays neutral Dynamic Geometry programs with automated proving features.), allowing the algorithmic with regard to jurisdictional claims in verification and derivation of geometric theorems, has renewed the interest in analyzing— published maps and institutional affil- and handling—some problems arising in the underlying interaction of the different pieces iations. involved in the GART protocols. In particular, we refer to the quite successful performance of such tools in the Dynamic Geometry program GeoGebra (https://www.geogebra.org/ download, accessed on 13 August 2021), freely available and with over one hundred million users all over the world, a fact that has reinforced and urged the need to address some pending problems. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. In this paper we begin by illustrating, through a collection of examples, the current This article is an open access article performance and ongoing development of GART in GeoGebra and its algebraic geometry distributed under the terms and background (see Sections2 and3). Then we reflect (Section4) on some (unexpected for conditions of the Creative Commons a standard user) problems that might arise, due to the interaction of the different steps Attribution (CC BY) license (https:// involved in the process, such as the user specific interpretation and introduction of the creativecommons.org/licenses/by/ geometric statement, the internal algebraic translation, and the algorithmic manipulation 4.0/). of polynomial ideals towards the final output (and again, its interpretation by the user). Mathematics 2021, 9, 1964. https://doi.org/10.3390/math9161964 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 1964 2 of 17 Finally, we focus on a particular kind of such problems (Section5), proposing a double approach that might help to avoid featuring counterintuitive conclusions due to unexpected degenerate instances of the given hypotheses (Section6). One is to deal with the saturation of the hypotheses ideal with respect to the ring of geometrically independent variables, thus allowing for the global consideration of the given statement over all non- degenerate components. The second is to reformulate the hypotheses ideal by treating the independent variables as invertible parameters, showing the specific advantages of this zero-dimensional case to analyze individually the truth of the statement over the different non-degenerate components. The paper concludes with some reflections on the potential application of these methods. 2. Automated Reasoning Tools in GeoGebra As described in [2,3], the standard version of GeoGebra includes now several commands: • For conjecturing some geometric property (e.g., “. these three lines visually “seem” to meet at one point. Is this actually true?. ”); • For rigorously confirming or denying a proposed conjecture (e.g., yielding an affir- mative answer after internally verifying, using Computer Algebra tools, that the equations describing the three selected lines have always a common solution); • For suggesting some complementary hypotheses for the truth of a given (initially false) statement (i.e., declaring some further restrictions in the geometric elements involved in the statement, that are required for its verification). 2.1. The Altitude Theorem Let us consider the classical “Geometric Mean” or “Altitude” Theorem attributed to Euclid (see [4], pp. 31–32). The traditional formulation states that in a right triangle, the length of the altitude on the hypotenuse is equal to the geometric mean of the two line segments it creates on the hypotenuse. However, suppose we “forget” about the condition to be a right triangle, yet we would like to remember the missing hypothesis, in order to rediscover this classic theorem. Then, we could start by conjecturing the truth of this theorem on a general triangle. Next, that of GART, the Prove command, declares (see Figure1) it is wrong. Figure 1. Left, GeoGebra is asked about the validity of the Altitude Theorem over an arbitrary triangle. Right, it declares it is false. Now, we ask for the requirements to place vertex C so that the Altitude Theorem holds. GeoGebra outputs two curves: a circle and a hyperbola (see Figure2, left image). Clearly, the first one corresponds to the right triangle case, that is, verifying that the sum of the two angles of the triangle at vertices A, B is 90 degrees; the second places C at some hyperbola. We have, thus, obtained two possible versions of the Altitude Theorem: the traditional one and the one corresponding to triangles ABC with C in the given hyperbola. As the equation of the hyperbola is given by GeoGebra, we can play a little with it (in a non-automatic way) Mathematics 2021, 9, 1964 3 of 17 concluding that such triangles are precisely those such that the difference of angles at A, B is also 90 degrees. Moreover, we can perform a specific construction of such triangles from a given pair of vertices A, B, placing point C at the hyperbola as follows (Figure2, right): first, choosing an arbitrary point G, considering the line f1 = GA, a perpendicular line through B, and its 0 0 symmetrical h with respect to the AB line. Then C is the intersection of f1, h . See [5] for details about this construction and generalized Altitude Theorem statement. Figure2, right image, shows how GeoGebra checks the truth of this extended version of the Altitude Theorem, yielding true as the Boolean Value corresponding to the question Prove (g2 == h · i), where g is the segment described by the altitude from vertex C and h, i are the segments DA, DB, respectively, going from the altitude feet D in the line AB, to the vertices A, B. Figure 2. Left, possible locus of C for the validity of the Altitude Theorem. Right, confirmation after placing C on the hyperbola. 2.2. A problem from a Spanish ‘Oposiciones’ Exam A different example of the performance of GeoGebra’s automated reasoning tools is displayed in Figure3. Given a quadrilateral ABCD, the midpoints E, F, G, H of the sides are built. Then GeoGebra is asked to find (left image) a Relation holding between the segments EF, GH. In the central image we can see that GeoGebra replies that the lengths of the two segments seem to be equal, at least numerically and for this particular assignment of the vertices of the quadrilateral. Thus, the user might conjecture that, perhaps, this fact holds in all generality, as a geometric statement. To check this conjecture, the user has to click on the More. button. Then GeoGebra launches internally the Prove command, yielding that it is indeed true (except for degenerate instances of the given construction) that the midpoints of the sides of the quadrilateral always form a parallelogram. However, the ability of the Relation command to find conjectures goes well beyond simple, school-like contexts. For example, in [6] it is shown how GeoGebra’s Relation is able to automatically conjecture and, then, prove, the alignment of some three points stated in Example 230 of the classical Chou’s benchmark [7]. Another, quite curious, case is depicted in Figure4, that shows two further examples of the powerful performance of the Relation command. The context is a problem proposed at a Spanish national exam ([8], p. 147) to become a certified mathematics teacher for secondary education. The candidates were given (on the exam sheet) a complicated figure that we can translate to a GeoGebra construction as follows (see Figure4). First, consider an equilateral triangle ABC, and build E, the midpoint of side AB, and D, the midpoint of AC.